Shortening the Hofer length of Hamiltonian circle actions

Abstract

A Hamiltonian circle action on a compact symplectic manifold is known to be a closed geodesic with respect to the Hofer metric on the group of Hamiltonian diffeomorphisms. If the momentum map attains its minimum or maximum at an isolated fixed point with isotropy weights not all equal to plus or minus one, then this closed geodesic can be deformed into a loop of shorter Hofer length. In this paper we give a lower bound for the possible amount of shortening, and we give a lower bound for the index (“number of independent shortening directions”). If the minimum or maximum is attained along a submanifold B, then we deform the circle action into a loop of shorter Hofer length whenever the isotropy weights have sufficiently large absolute values and the normal bundle of B is sufficiently un-twisted.